The QHE is characterized by two peculiar and interesting phenomena: the
quantization of the transverse conductivity, which is robust with respect to
small perturbations (like disorder), and the appearance of the
plateaux. The mathematical literature on the QHE is endless and cannot be
reported extensively here. For this reason, and at the cost of forgetting
important contributions, we will report below only the main contributions to the
subject which are related to the aim of the present work. The topological nature
of the quantization of the Hall conductivity is understood since 1981 with the
pioneering works [1]}, [2]}, but
its modern interpretation in terms of an index theory was completed
only in 1994, with the seminal paper [3]}. The existence
of the plateaux as the consequence of the presence of disorder was
mathematically proved in 1987 for the case of the Landau Hamiltonian
(continuous model) [4]}. An improved version of this result was
obtained only in more recent
years [5]}, [6]} thanks to the
development of new sophisticated mathematical tools like the multiscale
analysis. During the 80's, Bellissard had the brilliant intuition to combine
the noncommutative geometry (NGC), developed by Connes a few years earlier [7]}, [8]}, [9]}, with the study of the QHE. This
program eventually led to the groundbreaking paper [10]} of
1994 where the NGC of the QHE was
rigorously established. The main advantage of Bellissard's approach is
mainly related to the synthesis power of NGC. In fact, inside this new
framework, the complete phenomenology of the QHE (quantization and plateaux) can
be explained with the use of a unified language. The NGC program of the QHE has
been pushed forward by several other authors in last years
(e.g., [11]}, [12]}, [13]}, [14]}) and has been
extended with surprising success to the study of topological insulators
(e.g., [10]}, [16]}, [17]}).
